We discuss principality of prime ideals of finite algebraic number fields $L=K(\theta)$ over an algebraic number field $K ([K:\mathbb{ Q}]<\infty)$ defined by irreducible polynomials $f(x)\in \mathfrak{ O}_{ K}[x]$ and $f(\theta)=0$. Our main Theorem says that if a principal prime ideal $(\pi)\subset \mathfrak{ O}_{ K}$ is relatively prime to conductor $\mathfrak{ F} =\{ \alpha\in \mathfrak{ O}_{ L}|$ a principal ideal $(\alpha)$ of $\mathfrak{ O}_{ L}\subset \mathfrak{ O}_{ K}[\theta]\}$ and splits completely over $L$: $(\pi)\mathfrak{ O}_{ L}=\prod \mathfrak{ p}_{ i}$, then $\mathfrak{ p}_{ i}$ is a principal ideal of $\mathfrak{ O}_{ L}$ for all $i$, where $\mathfrak{ O}_{ L}= L \cap \overline{ \mathbb{ Z}}$ is integer ring of $L$. We use Jacobian Varieties of non-singular projective curve model of super elliptic curves $y^{ l}=f(x)$ to show the main Theorem, where $l$ is a large enough prime number which is relatively prime to degree of $f(x)$ and $(\pi)$.
Categories: math.NT